Wilson Central Terminal

Abstract

The expansion of the universe is currently accelerating, as first inferred by Efstathiou et al. (1990), Ostriker & Steinhardt (1995) and directly determined by Riess et al. (1998) and Perlmutter et al. (1999). Current constraints are consistent with a time independent equation-of-state of w = −1, which is to be expected when a constant vacuum energy density dominates.

But The Quantum Field

Theory prediction for the magnitude of this vacuum energy is very much larger than that inferred (Weinberg, 1989; Koksma & Prokopec, 2011). It is entirely possible that the cause of the expansion has an alternative explanation, with both the inclusion of a quantum scalar field and modified gravity theories able to reproduce an expansion history close to, but potentially deviating from, that of a cosmological constant and cold dark matter, ΛCDM; see Weinberg et al. (2013) for a review.

In this work I investigate the consistency of the VIMOS Public Extragalactic Redshift Survey (VIPERS) v7 census of the galaxy distribution at z = 0.8 with the expansion history and linear growth rate predicted by General Relativity (GR) when a Planck Collaboration et al. (2015) fiducial cosmology is assumed. To do so, I measure the optimally weighted redshift-space power spectrum (Feldman et al., 1994), which is anisotropic due to the coherent infall of galaxies towards overdensities and outflow from voids (Kaiser, 1987).

The Magnitude Of This

anisotropy can distinguish between modified theories of gravity as the convergence (divergence) rate of the velocity field depends on the effective strength of gravity on cosmological scales (Guzzo et al., 2008). This motivates measuring the linear growth rate rather than the background expansion, which is indistinguishable for a number of modified gravity theories. In Chapter 6 I place constraints of

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with the completed VIPERS v7 survey; the combination remains consistent with General Relativity at 95% confidence.

The Dependence Of The Errors On The

assumed priors will be investigated in future work. Further anisotropy is introduced by the Alcock-Paczy´nski effect (Alcock & Paczynski, 1979) – a distortion of the observed power spectrum due to the assumption of a fiducial cosmology differing from the true one.

These

two sources of anisotropy may be separated based on their distinct scale and angular dependence with sufficiently precise measurements. Doing so degrades

(1 + Z)Dah/C) To Be

simultaneously constrained. Galaxy redshift surveys may then directly compare both the background expansion and linear growth rate to the GR predictions (Ruiz & Huterer, 2015; Linder, 2016). I find the VIPERS v7 joint-posterior on (fσ8, FAP) shows no compelling deviation from the GR expectation although the sizeable errors reduce the significance of this conclusion.

In Chapter 4 I describe and outline corrections for the VIPERS spec- troscopic selection, which enable these constraints to be made. The VIPERS selection strategy is (projected) density dependent and may potentially bias measures of galaxy clustering. Throughout this work I present numerous tests of possible systematic biases, which are performed with the aid of realistic VIPERS mock catalogues. These also allow for accurate statistical error estimates to be made – by incorporating the sample variance due to both the finite volume and finite number density.

Chapter 5 details the development and testing of a new, rapid approach for the forward modelling of the power spectrum multipole moments obtained from a survey with an involved angular mask; this work has been accepted by the Monthly Notices of the Royal Astronomical Society as Wilson, Peacock, Taylor and de la Torre (2015).

An investigation of the necessary corrections for the VIPERS PDR-1 angular mask is recorded. This includes an original derivation for the integral constraint correction for a smoothed, joint-field estimate of ¯n(z) and a description of how the mask should be accounted for in light of the Alcock- Paczy´nski effect.

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Chapter 7 investigates the inclusion of a simple local overdensity transform: ‘clipping’ (Simpson et al., 2015) prior to the redshift-space distortions (RSD) analysis.

This tackles the root cause of non-linearity and potentially extends the validity of perturbation theory. Moreover, this marked clustering statistic potentially amplifies signatures of modified gravity (Lombriser et al., 2015; White, 2016) and, as a density-weighted two-point statistic, includes information not available to the power spectrum.

I show that a linear real-space power spectrum with a Kaiser factor and a Lorentzian damping yields a significant bias without clipping, but that this may be removed with a sufficiently strict transform; similar behaviour is observed for the VIPERS v7 dataset. Estimates of fσ8 for different thresholds are highly correlated due to the overlapping volume, but the bias for insufficient clipping can be calibrated and the correlation obtained using mock catalogues. A maximum likelihood value for the combined constraint of a number of thresholds is shown to achieve a ≃16% decrease in statistical error relative to the most precise single- threshold estimate. The results are encouraging to date but represent a work in progress; the final analysis will be submitted to Astronomy & Astrophysics as Wilson et al. (2016).

In addition to this, an original extension of the prediction for a clipped Gaussian field (Simpson et al., 2015) to a clipped lognormal field is presented. The results of tests of this model with a real-space cube populated according to the halo occupation distribution model (Zheng et al., 2005) are also provided.

Lay Summary

Cosmology asks many of the grandest questions in science, questioning how the universe in which we find ourselves came to be. Its remit extends to time and length scales that are beyond comprehension and yet, with concerted effort, a vast range of observations are now well described in a (mostly) consistent framework and often with principles derived from everyday experience.

While in Edinburgh, my own small part in this work has focused on a subfield of Cosmology named large-scale structure, which seeks to explain the spatial distribution of galaxies and harness this information in order to test the laws of gravity. To this end, I’ve had the pleasure of working with the VIPERS galaxy redshift survey, which provides a census of the positions and spectral properties of approximately 90,000 galaxies as the universe was when (at most) half its current age.

In our current understanding, the galaxy distribution is the product of initial density perturbations set by small-scale, quantum physics at least 13.4 billion years ago, which have since grown due to gravitational collapse and eventually formed stars and galaxies. In the interim, evidence suggests the young universe was extraordinarily hot, dense and bright – in fact, this light can be observed directly (at longer wavelengths) as the cosmic microwave background (Planck Collaboration et al., 2015). That an equally dense universe is not seen today is explained by a dilution of matter with time and is further evidenced by the observed recession of galaxies in our local neighbourhood.

While this current understanding consistently models a range of observa- tions, cosmologists have had to reluctantly accept various additions to physics for which there is no evidence for in the laboratory – the matter making up bicycles and books is seemingly only 4% of the total. This is a sorry state of ignorance and the current focus of cosmology is on learning more about the remainder.

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The observed trajectories of stars in galaxies and of galaxies in clusters suggest that a large percentage of this mass is present and simply does not emit light; it is therefore invisible to conventional optical astronomy. This hypothetical cold dark matter particle(s) is believed to make up 30% of the total mass. At most 0.5% can be contributed by neutrinos, which are a prime candidate as they have been detected, are known to possess mass and do not interact electromagnetically.

Perhaps the most likely candidate is a similar, weakly interacting particle, which is simply more elusive. The rate of galaxy recession is observed to be increasing with time, in direct conflict with the expected decrease due to the gravitational pull of baryonic and cold dark matter. This suggests that the remaining ≃70% of the energy content must be very different from normal baryonic matter. This is to simplify matters, as decades of quantum mechanics research suggests a likely candidate: the energy density of ‘empty’ space; this is a misnomer as truly empty space cannot be physically realised for any length of time. But the quantum prediction is very much different from the value required by cosmological observations – when General Relativity is assumed.

This Is One Instance, In A Recurring

theme, of a failure to combine our understanding of gravity with small-scale quantum physics. The problem is so large, one ‘solution’ is to speculate that this energy can’t possibly contribute and look for another source of the acceleration.

Numerous alternatives have been postulated, including the introduction of yet more previously undetected particles or modifying the theory of gravity on cosmological scales.

To proceed it seems legitimate to ensure that gravity is understood as well as we believe. This is especially the case on cosmological length scales, which have only recently become accessible. General Relativity has been extensively tested in the solar system in the past, with remarkable success. If cosmological tests can be performed in a manner that distinguishes between the predictions of alternative theories then so much the better; as the majority of alternatives are flexible enough to completely agree on the expected rate of galaxy recession, alternative tests gain prominence. Later chapters detail my work in ensuring the rate of gravitational collapse, as evidenced by the VIPERS galaxy distribution, is consistent with the predictions of General Relativity.

To do so, I exploit the effect of redshift-space distortions (Kaiser, 1987). As the attraction of gravity typically causes mass to coalesce, there is a large-scale infall of galaxies on the outskirts of a large cluster towards the cluster centre;

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this is shown in Fig. 2.1. This additional velocity causes the observed light from a given galaxy to be redder – due to the everyday Doppler effect (Doppler & Studnica 1903, Feynman 1963), when the galaxy lies between us and the cluster centre. By looking for a systematic change in the observed colour of galaxies on the outskirts of a cluster the rate of infall may be inferred. As this rate is predicted by General Relativity (once the total mass in the universe has been deduced by other means) the observed rate may be used to distinguish between modified theories of gravity (Guzzo et al., 2008).

Declaration

I declare that this thesis has been written solely by myself, that the work contained is my own except where otherwise stated and that no part has been submitted for any other degree or professional qualification.

The majority of Chapter 5 has been accepted by the Monthly Notices of the Royal Astronomical Society as Wilson, Peacock, Taylor and de la Torre (2015). Other than valuable proof reading of preliminary drafts, the work is my own. I’d like to thank Florian Beutler, Chris Blake, Ben Granett and Gigi Guzzo for providing helpful comments.

The work contained in Chapter 6 and Chapter 7 will be submitted to Astronomy and Astrophysics as Wilson et al. (2016).

Acknowledgements

This thesis would not have been possible without the aid of many people. Principally, my parents, Dorothy and Alfie Wilson, whose love, interest and sacrifice has been an everlasting support on which I could always depend.

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could never thank you enough and I hope this work does justice to your efforts. I would like to thank my supervisor, John Peacock, for being the guiding hand through what has occasionally seemed like choppy waters.

I Especially

appreciate your dedication, insight and ability to set the highest of standards to aspire to, all of which helped improved this thesis significantly. I owe a debt of gratitude to Sylvain de la Torre for his help – especially in my first year; without his obvious hard work for VIPERS, in particular with regard to the mocks on which this analysis is founded, the later chapters would look very different.

My thanks go to Andy Taylor and Catherine Heymans for their always open door. I’d like to thank Fergus Simpson for his originality, technical support and friendship. My appreciation extends to the entire VIPERS team, in particular to Gigi Guzzo, Ben Granett, Julien Bel, Stefano Rota and Andrea Pezzotta; The VIPERS conferences have always been enjoyable thanks to your approachability and willingness to offer helpful advice. My thanks go to Shaun Cole and Andy Lawrence for their enthusiastic and knowledgeable questioning during my viva.

Often the Observatory feels like a small place, but looking back it’s surprising to think of the number of people who have passed through. Although too many to name specifically, Ami Choi deserves special praise for being my longest friend here and for her support this year.

My Appreciation Goes To

Marco Lam for his friendship and continued patience as a flatmate. My time in Edinburgh has been all the better since inheriting an addiction to cycling from Becca, Chris, Sandy and David; relieving stress at five-a-side football; brewing beer with the expert guidance of Edouard, Esther & Jorge and dominating (for a spell) the Dagda pub quiz – thanks to Alex Hall and David. To the only source of stress I anticipate in the next few weeks, my chess nemeses: Fergus, Ami and Alex Hall, I’ll be working on my end game. Finally, my thoughts go to my long-suffering office mates, Shegy, Maria and Alex Amon in particular, who have had to endure my often foul mood in the past few months. Your company and continued support have made this all immeasurably easier.

In memory of my mother.

1.1

The cosmological principle and Hubble’s law .............................

1.2

A primer on Einstein gravity .................................................

1.3

The Robertson-Walker metric ................................................

1.3.1

Expansion dynamics...................................................

1.4

Physical separation from observables .......................................

1.4.1

Redshift ..................................................................

1.4.2

Distances................................................................. 10

1.5

Dark matter ...................................................................... 14

1.6

Dark energy and the cosmological constant ............................... 17

1.7

Modified gravity ................................................................. 21

1.8

Inflation............................................................................ 23

1.8.1

Deficiencies of classical cosmology.................................. 23

1.8.2

An early period of vacuum domination ........................... 24

1.9

Structure formation ............................................................. 27

1.9.1

The perturbed metric ................................................. 27

1.9.2

Newtonian structure formation ..................................... 30

1.9.3

Baryonic acoustic oscillations ....................................... 33

1.9.4

Cosmic microwave background...................................... 35 1.10 Thesis outline..................................................................... 37

2.1

Linear theory in the distant observer approximation.................... 39

2.2

Growth rate tests of modified gravity and dark energy models....... 44

2.3

Beyond the linear velocity field .............................................. 47

2.4

Symmetries of finite-angle RSD .............................................. 55

2.5

The Alcock-Paczy´nski effect .................................................. 56

2.6

Real-space correlations by deprojection .................................... 64

3.1

Mildly non-linear growth ...................................................... 69

3.1.1

Breakdown of linear theory .......................................... 69

3.1.2

The Zel’dovich approximation....................................... 70

3.1.3

Lognormal model....................................................... 71

3.2

Baryonic effects and galaxy bias ............................................. 73

3.3

Mock galaxy catalogues ........................................................ 75

3.4

Measures of clustering.......................................................... 76

3.4.1

Two-point estimators.................................................. 76

3.4.2

Practical power spectrum estimation.............................. 81

3.4.3

Maximum likelihood................................................... 89

3.4.4

Multipole covariance estimation .................................... 92

4.1

Survey details .................................................................... 95

4.1.1

Angular selection function ........................................... 97

4.1.2

Radial selection function ............................................. 102

4.1.3

Complementary VIPERS analyses ................................. 104

5.1

Introduction ...................................................................... 105

5.2

Power spectrum multipoles for a masked density field.................. 109

5.2.1

Practical details ........................................................ 112

5.2.2

A simple validity test.................................................. 114

5.3

Obtaining Qq(∆) with a pair counting approach......................... 114

5.4

VIPERS: an application to a realistic survey geometry ................ 117

5.4.1

A convergence test for VIPERS PDR-1........................... 119

5.5

Integral constraint correction ................................................. 120

5.5.1

Including a smoothed joint-field ¯n(z) estimate.................. 123

5.6

Masking and the Alcock-Paczy´nski effect.................................. 124

5.7

Complete impact of the VIPERS mask .................................... 124

5.8

Validity for finite-angle surveys .............................................. 125

5.9

Conclusions ....................................................................... 125 5.10 Derivation of eqn. (5.12)....................................................... 127

6

VIPERS: geometric and growth rate tests of gravity

6.1

Synopsis ........................................................................... 129

6.2

Modelling overview.............................................................. 131

6.3

A VIPERS v7 measurement of the growth rate .......................... 132

6.3.1

Measures of the power spectrum anisotropy ..................... 134

6.3.2

Derived constraints on fσ8(z)....................................... 136

6.3.3

Goodness-of-fit.......................................................... 143

6.4

Tests of systematics............................................................. 143

6.4.1

Potential biases due to the ¯n(z) estimate......................... 143

6.4.2

Potential biases due to the incompleteness correction ......... 145

6.4.3

Potential biases due to the assumed model ...................... 145

6.4.4

Potential biases due to the covariance estimation .............. 149

6.5

Statistical significance of the VIPERS v7 measurements............... 149

6.6

The geometric and growth rate degeneracy ............................... 152

6.6.1

Parameter degeneracies ............................................... 153

6.7

Future work....................................................................... 157

7.1

An effective weight definition ................................................. 167

7.2

Modelling the clipped power spectrum ..................................... 167

7.2.1

Spectral distortion of a clipped Gaussian field .................. 168

7.2.2

Spectral distortion of a clipped lognormal field ................. 170

7.3

Clipping the VIPERS v7 data................................................ 176

7.3.1

Modelling summary.................................................... 176

7.3.2

Results from the mocks............................................... 177

7.3.3

Results from the data ................................................. 180

7.4

Conclusions and future work.................................................. 185

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